Describe and calculate death strain at risk, expected death strain, actual death strain and mortality profit

Welcome to Mortality Profit and Loss!

In your CM1 journey so far, you have learned how to calculate premiums and reserves based on assumptions about how long people will live. But what happens if our assumptions don't match reality? That is exactly what this chapter is about!

We are going to look at how life insurance companies measure the "risk" they take on when someone dies and how they calculate whether they made a profit or a loss based on those deaths. Don't worry if this seems a bit abstract at first—we will break it down into four simple concepts: DSAR, EDS, ADS, and Mortality Profit.

1. Death Strain at Risk (DSAR)

Think of the Death Strain at Risk (DSAR) as the "financial hit" an insurance company takes at the exact moment a policyholder dies.

When a policyholder dies, the company must pay out the Sum Assured (S). However, the company hasn't "lost" the full amount because they have already been saving up a Reserve for that person. The DSAR is simply the difference between what they pay out and what they already had set aside.

The Formula:
For a policy in its \((t+1)\)-th year:
\(DSAR = S - {}_{t+1}V\)

Where:
\(S\) = The benefit payable on death.
\({}_{t+1}V\) = The reserve held at the end of the year (assuming the person survived).

Real-World Analogy:
Imagine you are saving up for a new laptop that costs £1,000. You have already saved £800 in your bank account. If your current laptop breaks today and you must buy the new one immediately, the "strain" on your extra finances is only £200, because you already had the £800 ready to go.

Quick Review:

• If the benefit is paid at the end of the year, the DSAR is \(S - {}_{t+1}V\).
• If the benefit is paid immediately on death, we usually adjust the formula to account for interest, but the core concept remains: Payout minus Reserve.

2. Expected Death Strain (EDS)

Before the year starts, the actuary looks at all the policyholders and asks: "Based on our mortality tables, how much do we expect to pay out in death strains this year?"

This is the Expected Death Strain (EDS). It is the probability of the person dying multiplied by the amount of "hit" the company takes if they do die (the DSAR).

The Formula:
\(EDS = q_{x+t} \times DSAR\)
\(EDS = q_{x+t} \times (S - {}_{t+1}V)\)

Key Concept:
The EDS is like a "budget" for deaths. We know we can't predict exactly who will die, but across thousands of policies, we expect a certain average cost.

3. Actual Death Strain (ADS)

At the end of the year, the actuary stops guessing and looks at the actual data. The Actual Death Strain (ADS) is what really happened.

How to calculate it:
• If the person is still alive at the end of the year: \(ADS = 0\)
• If the person died during the year: \(ADS = DSAR = S - {}_{t+1}V\)

For a large group of \(n\) identical policies, the total ADS is simply the number of actual deaths multiplied by the DSAR for one policy.

Did you know?
The ADS is always either a "full hit" or "zero" for a single person. It’s only when we look at a large group of people that the ADS starts to look like a smoother number that we can compare to our EDS.

4. Mortality Profit (Surplus)

Now for the most important part: did we make money or lose money? We calculate the Mortality Profit (also called Mortality Surplus) by comparing our "budget" (EDS) to our "actual cost" (ADS).

The Formula:
\(Mortality Profit = EDS - ADS\)

Understanding the Result:
• Positive Number: We have a Profit. This happens if \(EDS > ADS\) (fewer people died than we expected).
• Negative Number: We have a Loss. This happens if \(ADS > EDS\) (more people died than we expected).

Memory Aid: "E before A"
To remember the order, think of the alphabet. Expected comes before Actual. If you have more "Expected" money than "Actual" costs, you are in the green!

Common Mistake to Avoid:

Students often confuse Mortality Profit with Total Profit. Mortality profit only looks at the difference caused by death rates. It ignores differences in interest rates or expenses. Keep your focus strictly on the DSAR when working these problems!

5. Step-by-Step: Calculating Mortality Profit for a Group

If you are given a question about a group of policyholders, follow these steps:

1. Calculate the Reserve: Find \({}_{t+1}V\) for one policy.
2. Calculate the DSAR: Subtract the reserve from the Sum Assured (\(S - {}_{t+1}V\)).
3. Calculate Total EDS: Multiply the DSAR by the expected number of deaths (\(Number of policies \times q_{x+t} \times DSAR\)).
4. Calculate Total ADS: Multiply the DSAR by the actual number of deaths that occurred.
5. Subtract: \(Total EDS - Total ADS = Total Mortality Profit\).

Summary Table

DSAR: The "Gap" (\(S - V\))
EDS: The "Assumption" (\(q \times DSAR\))
ADS: The "Reality" (Actual Deaths \(\times DSAR\))
Mortality Profit: Expected minus Actual (\(EDS - ADS\))

Encouragement: You've got this! Mortality profit is just a way of checking our work. If you can calculate a reserve and you know your \(q_x\) values, you are 90% of the way there!